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Publications [#348659] of Robert Bryant

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Papers Published

  1. Bryant, RL; Clelland, JN, Flat metrics with a prescribed derived coframing, Symmetry, Integrability and Geometry: Methods and Applications, vol. 16 (January, 2020) [doi]
    (last updated on 2021/05/10)

    Abstract:
    The following problem is addressed: A 3-manifold M is endowed with a triple Ω =(Ω , Ω , Ω ) of closed 2-forms. One wants to construct a coframing ω =(ω , ω , ω ) of M such that, first, dω = Ω for i = 1, 2, 3, and, second, the Riemannian metric g = ( ) + (ω ) + (ω ) be flat. We show that, in the ‘nonsingular case’, i.e., when the three 2-forms Ω p span at least a 2-dimensional subspace of Λ (T *M) and are real-analytic in some p-centered coordinates, this problem is always solvable on a neighborhood of p (Formula Presented) M, with the general solution ω depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution ω can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when Ω , Ω , Ω are scalar multiples of a single 2-form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied. 1 2 3 1 2 3 i i 1 2 2 2 3 2 i 2 1 2 3 ω p

 

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